Please try it without /etc/X11/xorg.conf.d/20-synaptics.conf and synclient, is it dead alltogether, does it behave like a normal mouse, what about tapping and scrolling - what does /var/log/Xorg.0.log say?

My own tests, however with much older hardware, have been successful with SynPS/2 Synaptics TouchPad and AlpsPS/2 ALPS GlidePoint.

I'll try without them (synclient -l was only used to check current settings), however I would be grateful if You would point me how to make tapping acting like a 3rd mouse button (middle click) in a up to date way? Not to mention the fact that it's a little strange for me that these work with previous kernel (even today, I have one still installed).

So far I have yet another strange fact, if I do rmmod psmouse and after that do insmod /lib/modules/3.6-0.slh.1-aptosid-amd64/kernel/drivers/input/mouse/psmouse.ko, then xinput list, again shows it as SynPS/2 Synaptics TouchPad and my config works ok.

That would mean the touchpad gets correctly detected as a touchpad, the mouse emulation fallback (predominantly supported by alps devices) wouldn't support scrolling at all (just mousepointer && left click), suggesting 'just' a configuration issue. Unfortunately multi-touch is not supported by the touchpads I have access to, so I can't help a lot with specifics in this regard.

The module loading differences are a bit weird, but for that we'd need further logging output.

Some update,
without the .conf file in /etc/X11/xorg.conf.d, after first reboot xinput showed SynPS/2 Synaptics TouchPad.
After second reboot touchpad again was shown as ImPS/2 Synaptics TouchPad - however I managed to spot a suspicious entry in the dmesg output:

As far as the touchpad goes, I would just like to have an option to configure tapping the touchpad as a "middle" mouse button click, the multitouch options such as double finger scrolling are not really needed (and they seem to come and go depending on the version o xserver packages and/or kernel combined with my hardware).